Maple syrup is produced from maple sap, which contains about $4\mathrm{.}0\frac{w}{w}\%$ sugar, by staged evaporation of the water from the sap, leaving behind an increasingly concentrated sugar solution.

a$)$ If a cascade of flash evaporators is used to pretreat $200k\frac{g}{h}$ of maple syrup prior to final boiling into syrup, how many evaporators are needed to increase its sugar concentration to $5\mathrm{.}0\frac{w}{w}\%?$ Each evaporator reduces pressure by $80\%$ from the previous one, starting with a high$-$pressure sap feed at $12\mathrm{.}0$ bar, and each evaporator operates at the saturation temperature of its feed. The vapour stream is vented, and the liquid stream proceeds to the next evaporator.

b$)$ The sap is piped to each evaporator along a $3/4"$ SDR $9$ PEX line, which enters at the bottom and exits at the top of the evaporator. Determine the height $($i$.$e$.$ difference between the heights of the inlet and outlet lines$)$ of the final evaporator in the cascade, assuming a constant sap density of $1070k\frac{g}{m^3}$ and no frictional losses.